7  Cauchy’s theorem

Definition 7.1 Let \(D\subset\C\) and \(\ga_0, \ga_1\colon[a,b]\to D\) curves in \(D.\)

1. A homotopy in \(D\) between paths \(\ga_0, \ga_1\) is a continuous map \[\Ga\colon[0,1]\t[a,b]\longra D, (s,t)\longmapsto\Ga_s(t),\] such that \(\Ga_0(t)=\ga_0(t)\) and \(\Ga_1(t)=\ga_1(t)\) for all \(t\in[a,b].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic paths in \(D.\)

   If, additionally, \(\Ga_s(a)=p\) and \(\Ga_s(b)=q\) are constant in \(s\in[0,1],\) we call \(\Ga\) a path homotopy in \(D\) and \(\ga_0, \ga_1\) path-homotopic in \(D\).

2. If \(\ga_0, \ga_1\) are loops, a homotopy of loops in \(D\) is a homotopy \(\Ga\) in \(D\) with the additional property that \(\Ga_s\) is a loop for each \(s\in[0,1].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic loops in \(D.\)

   A loop is null-homotopic in \(D\) if there is a homotopy of loops in \(D\) to the constant loop.

Covered in lectures. Check back once the chapter is concluded.

Remark 7.1. In the following we will also suppose that \(\Ga\) is piecewise C¹, meaning there exist subdivisions \[0=s_0<s_1<\cdots<s_m=1,\qquad a=t_0<t_1<\cdots<t_n=b\]

such that each restriction \(\Ga|_{[s_{j-1},s_j]\t[t_{k-1},t_k]}\) is continuously differentiable.

Example 7.1  

Covered in lectures. Check back once the chapter is concluded.

Figure 7.1: Augustin Louis, Baron Cauchy. Lithograph by J. Boilly, 1821. Wellcome Collection. Public Domain Mark

Theorem 7.1 (Cauchy’s theorem) Let \(f\colon U\to \C\) be a holomorphic function on an open set. Let \(\ga_0, \ga_1\) be piecewise C¹ curves in \(U\) that are path-homotopic in \(U.\) Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]

Proof.

Covered in lectures. Check back once the chapter is concluded.

Theorem 7.2 Let \(\ga_0, \ga_1\) be piecewise C¹ loops in \(U\) that are freely homotopic in \(U.\) Suppose \(f\colon U\to\C\) is a holomorphic function. Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]In particular, if \(\ga\) is a loop that is homotopic in \(U\) to the constant loop, then\[\int_\ga f(z)dz=0. \tag{7.1}\]

Proof.

Covered in lectures. Check back once the chapter is concluded.

Questions for further discussion

Covered in lectures. Check back once the chapter is concluded.

• Give a counterexample to Cauchy’s theorem when (i) \(\ga\) is not null-homotopic (ii) \(f(z)\) is not holomorphic

• Does Cauchy’s theorem hold for \(f(z)=\ol{z}\)?

7.1 Exercises

Exercise 7.1

1. Sketch the curves \[\begin{align*} \ga_0\colon[-1,1]&\longra\C,&\ga_0(t)&=t,\\ \ga_1\colon[-1,1]&\longra\C,&\ga_1(t)&=e^{i\pi\frac{1-t}{2}} \end{align*}\] and show that they are path-homotopic.

   2. Let \(0<r_0<r_1\) and \(z_0\in\C.\) Prove that the loops \(\partial D_{r_0}(z_0)\) and \(\partial D_{r_1}(z_0)\) are freely homotopic in the closed annulus \(\ol A_{r_0,r_1}(z_0).\)

Exercise 7.2

Let \(\al+i\be\in\C.\) Determine \[\int_a^b e^{(\al+i\be)t} dt\] to compute \[\int_a^b e^{\al t}\cos(\be t)dt.\]

Exercise 7.3

Let \(\C^-=\C\setminus(-\iy,0]\) be the slit plane.

1. Show that any two points in \(\C^-\) may be connected by a path in \(\C^-.\) Hence \(\C^-\) is path-connected.
2. Show that every closed curve \(\ga\colon[a,b]\to\C^-\) is null-homotopic in \(\C^-\) by finding a homotopy \[\Ga\colon[0,1]\t[a,b]\longra\C^-, (s,t)\longmapsto\Ga_s(t)\] satisfying \(\Ga_0(t)=\ga(t)\) and \(\Ga_1(t)=1\) for all \(t\in[a,b].\) Hence \(\C^-\) is simply-connected.
3. Prove the analogues of a. and b. for a disk \(D_r(z_0).\)
4. Use Cauchy’s Theorem to prove that the punctured plane \(\C^\t=\C\setminus\{0\}\) is not simply-connected. That is, there exists a closed curve in \(\C^\t\) that is not null-homotopic in \(\C^\t.\)

Exercise 7.4

Let \(0<b<1.\)

1. Using the geometric series, find the power series expansion \[\frac{1}{z-1/b}=\sum_{n=0}^\iy a_n(z-b)^n\] with center \(z_0=b\) and determine the radius of convergence \(\rho.\)
2. Use a. to show that for all \(0<r<\rho\) \[\int_{\partial D_r(b)}\frac{dz}{(z-b)(z-1/b)}=\frac{2\pi i}{b-1/b}.\]
3. Use c. to compute \[\int_0^{2\pi}\frac{dt}{1-2b\cos(t)+b^2}.\]

Exercise 7.5

Let \(\ga\colon[0,1]\to D\) be a curve in \(D\subset\C\) and let \(-\ga\colon[0,1]\to D,\) \((-\ga)(t)=\ga(1-t)\) be the opposite curve. Prove that \(\ga\ast(-\ga)\) is path-homotopic to a constant loop.

Dowson, H. R. 1979. “Serge Lang, Complex Analysis (Addison-Wesley, 1977), Xi + 321 Pp., £1200.” Proceedings of the Edinburgh Mathematical Society 22 (1): 65–65. https://doi.org/10.1017/S0013091500027838.

Freitag, E., and R. Busam. 2009. Complex Analysis. Universitext. Springer Berlin Heidelberg. https://books.google.co.uk/books?id=3xBpS-ZKlgsC.

Howie, J. M. 2012. Complex Analysis. Springer Undergraduate Mathematics Series. Springer London. https://books.google.co.uk/books?id=0FZDBAAAQBAJ.

Jameson, G. J. O. 1987. “H. A. Priestley, Introduction to Complex Analysis (Oxford University Press, 1985), 197 Pp., £8.50.” Proceedings of the Edinburgh Mathematical Society 30 (2): 325–26. https://doi.org/10.1017/S0013091500028406.

Remmert, R. 1991. Theory of Complex Functions. Graduate Texts in Mathematics. Springer New York. https://books.google.co.uk/books?id=uP8SF4jf7GEC.

Thomson, B. S., J. B. Bruckner, and A. M. Bruckner. 2001. Elementary Real Analysis. Prentice-Hall. https://books.google.co.uk/books?id=6l_E9OTFaK0C.